Kondo Destruction and Quantum Criticality in Kondo Lattice Systems

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Journal of the Physical Society of Japan 83, 061005 (2014) Special Topics http://dx.doi.org/10.7566/JPSJ.83.061005 Advances in Physics of Strongly Correlated Electron Systems

Qimiao Si1+, Jedediah H. Pixley1, Emilian Nica1, Seiji J. Yamamoto1, Pallab Goswami2, Rong Yu3, and Stefan Kirchner4,5

1Department of Physics and Astronomy, Rice University, Houston, TX 77005, U.S.A. 2National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310, U.S.A. 3Department of Physics, Renmin University of China, Beijing 100872, China 4Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany 5Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany (Received December 3, 2013; accepted December 26, 2013; published online May 9, 2014) Considerable efforts have been made in recent years to theoretically understand quantum phase transitions in Kondo lattice systems. A particular focus is on Kondo destruction, which leads to quantum criticality that goes beyond the Landau framework of order-parameter ﬂuctuations. This unconventional quantum criticality has provided an understanding of the unusual dynamical scaling observed experimentally. It also predicted a sudden jump of the Fermi surface and an extra (Kondo destruction) energy scale, both of which have been veriﬁed by systematic experiments. Considerations of Kondo destruction have in addition yielded a global phase diagram, which has motivated the current interest in heavy fermion materials with variable dimensionality or geometrical frustration. Here we summarize these developments, and discuss some of the ongoing work and open issues. We also consider the implications of these results for superconductivity. Finally, we address the effect of spin–orbit coupling on the global phase diagram, suggest that SmB6 under pressure may display unconventional superconductivity in the transition regime between a Kondo insulator phase and an antiferroamgnetic metal phase, and argue that the interfaces of heavy-fermion heterostructures will provide a fertile setting to explore topological properties of both Kondo insulators and heavy- fermion superconductors.

1. Introduction Quantum criticality is currently being studied in a wide variety of strongly correlated electron systems. It provides a mechanism for both non-Fermi liquid excitations and unconventional superconductivity. Heavy fermion metals represent a prototype system to study the nature of quantum criticality, as well as the novel phases that emerge in the vicinity of a quantum critical point (QCP).1,2) Over the past decade, Kondo lattice systems have provided a setting for extensive theoretical analysis of quantum phase transitions between ordered antiferromagnetic (AF) and paramagnetic ground states. Various studies have revealed a class of unconventional QCPs that goes beyond the Landau framework of order-parameter fluctuations. This local Fig. 1. (Color online) Quantum critical behavior in the generic phase diagram of temperature and a non-thermal control parameter. quantum criticality incorporates the physics of Kondo destruction. Considerations of unconventional quantum criticality have naturally led to the question of the role of symmetry of the Hamiltonian, and is therefore a magneti- Kondo destruction in the emergent phases. Consequently, a cally-disordered state. global phase diagram has recently been proposed. In general, the ratio of such competing interactions In this article, we give a perspective on this subject and specifies a control parameter, which tunes the system from discuss the recent developments. We also point out several one ground state to another through a quantum phase outstanding issues and some new avenues for future studies. transition. A typical case is illustrated in Fig. 1, where the quantum phase transition goes from an ordered state to a 2. Quantum Criticality disordered one. When it is continuous, the transition occurs at A quantum many-body Hamiltonian may contain terms a QCP. that lead to competing ground states. A textbook example3,4) In the Landau framework, the phases are distinguished is the problem of a chain of Ising spins, containing both a by an order parameter, which characterizes the spontaneous nearest-neighbor ferromagnetic exchange interaction between symmetry breaking. The quantum criticality is then described the spins and a magnetic field applied along a transverse in terms of d þ z-dimensional fluctuations of the order direction. The exchange interaction favors a ground state in parameter in space and time. Here, d is the spatial dimension which all the spins are aligned, which spontaneously breaks a and z is the dynamic exponent. global Z2 symmetry and yields the familiar ferromagnetic For weak metallic antiferromagnets, the magnetization order. The transverse field, on the other hand, prefers a associated with the ordering wavevector characterizes a spin- ground state in which all the spins point along the transverse density-wave (SDW) order. The QCP separates the SDW direction; this state does not spontaneously break any phase from a paramagnetic Fermi liquid state. The collective 061005-1 ©2014 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Rice University on 06/01/17

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is lowered. The RG flow is towards a strong-coupling fixed point, which controls the physics below a bare Kondo energy 0 1 expð1 Þ scale: TK 0 =0JK , where 0 is the density of states of the conduction electrons at the Fermi energy. At the fixed point, the local moment and the spins of the conduction electrons are locked into an entangled singlet state: 1 j i¼ ðj"i j#i j#i j"i Þ ð3Þ Kondo singlet 2 f c,FS f c,FS ; δ j i where c,FS represents a linear combination of the conduction-electron states close to the Fermi energy. This singlet ground state supports a resonance in the low- energy electronic excitation spectrum. The Kondo resonance can clearly be seen in an analysis of the strong-coupling limit, when JK is taken to be larger than the bandwidth of the conduction electrons and, for the physical case of small JK, has been readily described in terms of a slave boson method.14) The Kondo coupling is converted into an effective δ hybridization, b , between an emergent fermion f and the conduction electrons. Fig. 2. (Color online) Local quantum criticality (top panel) and the ¤ corresponding -dependence of the quasiparticle spectral weights zS and 3.2 Kondo lattice and heavy Fermi liquid zL, respectively for small and large Fermi surfaces (bottom panel). Here, 0 T =I is the control parameter, and T0 marks the initial onset of the In stoichiometric heavy fermion compounds containing, K fi 4 Kondo screening process; TN and TFL are respectively the Néel and Fermi- e.g., Ce or Yb elements, the partially- lled f electrons are liquid temperatures. Eloc characterizes the Kondo destruction, separating the strongly correlated. They behave as a lattice of effective spin- part of the phase diagram where the system flows towards a Kondo-singlet 1/2 local moments, which describe the magnetic degrees of fl ground state from that where the ow is towards a Kondo-destroyed ground freedom of the lowest Kramers-doublet atomic levels. This state. The bottom panel also illustrates the small (left) and large (right) Fermi surfaces, and the fluctuating Fermi surfaces (middle) associated with the yields a Kondo lattice Hamiltonian: X X QCP. ¼ þ S S þ S sc ð4Þ HKL H0 Iij i j JK i i : ij i 4 fluctuations are described in terms of a theory of order- The Kondo coupling JK is antiferromagnetic and we will 5) parameter fluctuations. focus on an antiferromagnetic RKKY interaction, Iij > 0. In heavy fermion metals, QCPs between an AF phase and At high energies, the local moments are essentially a paramagnetic heavy-fermion state have been observed in decoupled, and Eq. (2) would continue to apply, signifying a number of compounds.1,2) The local quantum criticality the initial development of Kondo screening process. What (Fig. 2) has new critical modes associated with the happens in the ground state, however, will depend on the destruction of the Kondo effect, in addition to the fluctuations competition between the Kondo and RKKY interactions. of the AF order parameter.6,7) It has provided an under- Consider first the regime where the Kondo effect 0 standing of unusual dynamical scaling properties observed dominates, with TK being much larger than the RKKY in quantum critical heavy fermion metals,8,9) and made interaction. The physics of this regime can be inferred by predictions regarding the evolution of Fermi surfaces and taking the bare Kondo coupling JK to be greater than the emergence of new energy scales that have been verified by bandwidth W of the conduction electrons.15–17) The Fermi 10–13) subsequent experiments in YbRh2Si2 and CeRhIn5. surface will be large, enclosing 1 þ x electrons per unit cell. When JK=W is reduced to being considerably smaller than 1, 3. From the Kondo Effect to its Destruction 0 while keeping I=TK small, continuity dictates that the 3.1 Kondo effect entangled Kondo singlet state still characterize the ground The Kondo effect was originally studied in the context of a state, and the Fermi surface will remain large. This can be single-impurity Kondo model: seen, microscopically, through the slave-boson ap- 18–20) c proach. The Kondo resonance in the excitation spectrum HKondo ¼ H0 þ JKS s0: ð1Þ P appears as a pole in the conduction-electron self-energy: y c y Here, H0 ¼ k "kck ck, s0 ¼ c0 ð0 =2Þc00 , with 2 ; y ðb Þ denoting a vector of Pauli matrices, and c creates an ðk Þ¼ ð5Þ 0 ;! ; electron of spin · at the impurity site 0; the Kondo coupling ! "f JK is antiferromagnetic (JK > 0). The renormalization-group where the self energy is defined through the Dyson equation: 14) 1 (RG) beta-function is, to quadratic order: Gcðk;!Þ¼½! "k ðk;!Þ . The conduction-electron Green’s function now has two poles, at energies dJK ð Þ¼ 2 ð2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi JK JK: ¼ð1 2Þ½ þ ð Þ2 þ 4ð Þ2 ð6Þ dl Ek = "k "f "k "f b ; The positive sign of the beta function implies that the effective which describe the heavy-fermion bands. As illustrated in Kondo coupling is marginally relevant, growing as the energy Fig. 3, the nonzero b describing the Kondo resonances is 061005-2 ©2014 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Rice University on 06/01/17

J. Phys. Soc. Jpn. 83, 061005 (2014) Special Topics Q. Si et al.

to the formation of the Kondo singlets. The essential question is whether this effect is sufﬁciently strong to destroy the amplitude of the static singlet, b, and correspondingly drive ¼ the energy scale (Eloc) or temperature scale (Tloc Eloc=kB, where kB is the Boltzmann constant) to zero. Such Kondo destruction was discussed early on in RG analyses of models containing Kondo-lattice-type effects.27–30) A systematic study became available in Refs. 6 and 31 which provided an understanding of the anomalous spin dynamics measured8) in the quantum critical heavy fermion metal CeCu5.9Au0.1. These studies also predicted the collapse of the Kondo- destruction energy scale Eloc, a sudden jump of the Fermi surface across the QCP, and the critical nature of the quasiparticles on the whole Fermi surface at the QCP (cf. Fig. 2). This approach was connected to some general theoretical considerations.7) The Kondo destruction has since Fig. 3. (Color online) (Left) The energy dispersion of the conduction- been studied using various other methods, including a electron band. The Fermi surface is small in that it only involves the x 32,33) conduction electrons per unit cell. (Right) The bands of the hybridized heavy fermionic slave-particle approach and dynamical mean 34) Fermi liquid. A Kondo/hybridization gap separates the two bands. The Fermi ﬁeld theory. The critical quasiparticles on the entire Fermi surface is large in that it also counts the local moments. Without a loss of surface have also been considered using a self-consistent generality, we have taken 0

4. Kondo Destruction and Quantum Criticality of directly responsible for a large Fermi surface. The quasi- Antiferromagnetic Heavy Fermion Metals 2 particle residue goes as zL /ðb Þ (Fig. 2). An AF QCP is expected when the control parameter ¤ In addition to the Kondo coupling between the local becomes sufficiently small, i.e., when the RKKY interaction moments and conduction electrons, the Kondo-lattice is large enough. One microscopic approach that has been Hamiltonian also contains an RKKY interaction among the playing an important role is the extended dynamical mean- local moments. In Eq. (4), this has been explicitly incorpo- field theory (EDMFT).27,36,37) rated. The RKKY interaction I represents an energy scale that 0 21,22) fi competes against TK, the Kondo scale. We de ne the 4.1 EDMFT approach 0 ratio of the two energy scales, TK=I, to be the tuning In the EDMFT approach, the fate of the Kondo effect is parameter. studied through the Bose–Fermi Kondo model, X Historically, the beginning of the heavy-fermion field H ¼ H þ w y focused attention on the Fermi liquid behavior highlighted by imp Kondo p p p p a large carrier mass, as well as the exploration of unconven- X þ S ð þ y Þ ð7Þ tional superconductivity. It was gradually realized that the g p p ; Fermi liquid description can break down.23,24) In the modern p ð Þ¼ era, there is now wide recognition that quantum criticality Palong with the self-consistencyP equations loc ! underlies the non-Fermi liquid behavior in many, if not all, q ðq;!Þ, and Glocð!Þ¼ k Gðk;!Þ. Associated with 1 heavy-fermion systems. Eq. (7) are the Dyson equations: Mð!Þ¼0 ð!Þþ 1 ð Þ ð Þ¼ 1ð Þ1 ð Þ =loc ! andP ! G0 ! =Gloc ! ,P where 1ð Þ¼ 2 2 ð 2 2 Þ ð Þ¼ 1 3.3 Quantum criticality: From Landau approach to Kondo 0 ! g p wp= ! wp and G0 ! p = destruction ð! EpÞ. The self-consistency equations manifest spatial To study quantum criticality in heavy-fermion metals, dimensionality of the magnetic fluctuationsP through the form one would normally assume that the Kondo effect remains of the RKKY density of states IðxÞ q ðx IqÞ.Two intact across the QCP. The ordered state is then an SDW dimensional magnetic fluctuations correspond to a I ðxÞ and the QCP follows the Landau approach introduced by which is nonzero at the lower edge, as typified by the case: 5,25,26) Hertz for weak antiferromagnets. Because the dy- IðxÞ¼ð1=2IÞðI jxjÞ, where © is the Heaviside step namic exponent in this approach is z ¼ 2, the effective function. On the other hand, three dimensional magnetic dimension d þ z is larger than or equal to 4 (the upper fluctuations are represented by a IðxÞ with a square-root 4 critical dimension of the theory). Therefore, the fixed form near thep lowerffiffiffiffiffiffiffiffiffiffiffiffiffiffi edge, as given by the example of 2 2 2 point is Gaussian. IðxÞ¼ð2=I Þ I x ðI jxjÞ. The search for beyond-Landau quantum criticality has focused on the phenomenon of Kondo destruction, from 4.2 Kondo destruction which emerges the inherently quantum modes that do not In the EDMFT approach, the dynamical magnetic connect with any spontaneously broken symmetry. correlations of the local moments influence the Kondo To study the Kondo destruction, it is important to analyze effect through the bosonic bath. Irrespective of the spatial the dynamical competition between the RKKY and Kondo dimensionality, the bosonic bath has a softened spectrum near interactions. The RKKY interaction induces dynamical the magnetic QCP. Correspondingly, it causes an enhanced correlations among the local moments, which are detrimental suppression of the Kondo effect. This effect has been studied 061005-3 ©2014 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Rice University on 06/01/17

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5. Global Phase Diagram JK Kondo destruction and the associated Fermi-surface change represent physics that goes beyond the Landau framework. Considerations of new phases that reﬂect this physics have led to a global phase diagram for the AF Kondo-lattice systems.47–49) This phase diagram was ﬁrst developed based on theoretical studies showing the stability 47,50–52) of the AFS phase. This is an AF phase with Kondo g destruction and an associated small Fermi surface.

Fig. 4. (Color online) RG flow of the Bose–Fermi Kondo model [with 5.1 Kondo destruction inside antiferromagnetic order: >0; cf. Eq. (8)], showing the strong-coupling Kondo fixed point and its QNL·M approach destruction. Can the Kondo effect be destroyed inside the AF ordered phase? To address this issue, Ref. 50 considered the Kondo lattice model with SU(2) symmetry, and in the parameter extensively, as in Ref. 38. It can be most clearly seen through limit of JK I W. (The case with Ising anisotropy is an RG approach of the Bose–Fermi Kondo problem that simpler: because the AF-ordered phase has a spin gap, JK is utilizes an ¥-expansion, where ¥ is defined through the irrelevant and the AFS phase will be stable.) We consider the ¼ 0 Pdeviation of the bosonic spectrum from the Ohmic form: reference limit JK to be the local moments in a collinear ð Þ 1 p ! wp ! . The RG equations of the Kondo AF order, described by a quantum non-linear sigma model 2 problem, Eq. (2), now take, to quadratic order in JK and g , (QNL·M) and, separately, the conduction electron band. The the following form:6,28–30,39) QNL·M53,54) takes the form: 2 Z ðJKÞ¼JKðJK g Þ; d 2 2 2 SQNLM ¼ðc=2gÞ d xd½ðrnÞ þ c ð@nÞ ; ð11Þ ðgÞ¼gð=2 g2Þ: ð8Þ The RG equations yield a Kondo-destruction critical point, where c is the spin-wave velocity and g measures the amount as shown in Fig. 4. It should be stressed that the zero- of quantum fluctuations, which grows as the amount of temperature local spin susceptibility has the form: frustration is increased (such as via tuning the ratio of next nearest neighbor to nearest neighbor spin–spin interactions ðÞ : ð9Þ loc on a square lattice). Importantly, the critical exponent shown above is valid to The case of the AF zone boundary not intersecting the infinite orders in ¥ (Ref. 39). Fermi surface of the conduction electrons allows an asymptotically exact analysis. Expressed in terms of the 4.3 Local quantum critical point n field of the QNL·M, which represents the staggered The EDMFT equations have been studied in some detail magnetization, the Kondo coupling takes the following form in a number of analytical and numerical studies.6,31,40–44) at low energies: Z Irrespective of the spatial dimensionality, the weakening d of the Kondo effect is seen through the reduction of the SK ¼ K d xdsc n @n: ð12Þ Eloc scale as we approach the QCP from the paramagnetic side. While the RG procedure55) appropriate for combined gapless 56) fi Eloc vanishes at the QCP for two-dimensional magnetic fermionic and bosonic elds is in general very involved, fluctuations. The dynamical spin susceptibility satisfies the the situation simplifies here because the QNL·M has a following dynamical scaling: dynamic exponent z ¼ 1. The kinematics involved in the RG 1 approach is illustrated in Fig. 5. The resulting RG equation is: ðq Þ¼ ð10Þ ;! : fðqÞþAði!Þ Mð!=TÞ ðK Þ¼0 ð13Þ

The exponent ¡ is found to be near to 0.75 (between 0.72 and In other words, K is exactly marginal. The Kondo coupling 0.83 derived from different approaches).40,43,44) does not grow, and there is no Kondo singlet formation in the ﬂ For three-dimensional magnetic uctuations, Eloc is ground state; i.e., the AF phase has a Kondo destruction. This reduced but remains non-zero at the QCP. It however establishes the stability of the AFS phase. A large-N analysis terminates inside the ordered portion of the phase diagram of the low-energy excitations was also carried out in Ref. 50, (see below). yielding a self-energy for the conduction electrons: In both cases, the zero-temperature transition is second- ðk;!Þ/!d: ð14Þ order when the effective RKKY interaction appears in the same form on both sides of the transition.45,46) It is important This self-energy lacks a pole, which is to be contrasted with to stress one effect that is crucial for both the stability of the Eq. (5). In other words, the Kondo resonance is absent, and Kondo-destroyed AF phase as well as the second-order the Fermi surface is small. nature of the QCP: a dynamical Kondo effect still operates in the Kond-destroyed AF phase. We will expound on this point 5.2 Global phase diagram in Sect. 6. The stability of the AFS phase raises the question, what are 061005-4 ©2014 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Rice University on 06/01/17

J. Phys. Soc. Jpn. 83, 061005 (2014) Special Topics Q. Si et al.

5.3 Specific cases and multiplicity of tuning parameters Our discussion so far is very general. To make further progress, it is important to consider the specific cases as well as the specific realizations of the parameter G. One case which is amenable to concrete calculations is the Ising-anisotropic Kondo problem in the presence of a transverse magnetic field. As already mentioned in the introduction, the transverse field introduces quantum fluctuations for the local moments, and provides a means to tune the G axis. In an EDMFT study, this leads to a Bose–Fermi Kondo model, with Ising anisotropy and in the presence of a transverse field, which is supplemented by self-consistency conditions. The transverse-field Bose–Fermi Kondo model 57) Fig. 5. (Color online) Different kinematics in the scaling of the bosonic per se has recently been studied in detail. The calculations and fermionic sectors. Figure adapted from Ref. 55. have been carried out using a version of the numerical renormalization group method,58) and a line of Kondo- destruction fixed points was identified. Another setting for concrete calculations is the spin- symmetric Kondo lattice model on the Shastry–Sutherland lattice. The parameter J2=J1 — the ratio of the exchange interaction on a diagonal bond to that on the nearest-neighbor bond — measures the degree of frustration and is defined as G. A key advantage is that, at large G and JK ¼ 0 the ground state of the local-moment only model is known exactly to be a valence-bond solid.59) A large-N-based calculation60) yields a phase diagram that is reminiscent of Fig. 6 when the conduction electrons are away from half-filling.

Fig. 6. (Color online) The global phase diagram of the AF Kondo 5.4 Berry phase and the topological defects of Néel order lattice.47,48) This T ¼ 0 phase diagram involves a frustration axis (G) and an axis that tunes the Kondo coupling (JK). PL and PS are paramagnetic Considerations of the global phase diagram also opens up phases whose Fermi surfaces are respectively large and small (i.e., with or the study of the heavy-fermion state based on the Berry phase without Kondo resonances). AFL and AFS are their AF counterparts. and topological defects of local-moment magnetism. This was recently studied in the QNL·M representation of the spin one-half Kondo lattice model on a honeycomb lattice at half the different possible routes to suppress AF order and tune filling61) (see also Ref. 62 for the 1D case). It has been shown the system from the AFS phase towards the paramagnetic that the skyrmion defects of the antiferromagnetic order heavy-fermion state (PL phase)? The routes are illustrated in parameter host a number of competing states. In addition the global phase diagram, Fig. 6. This zero-temperature to the spin Peierls, charge and current density wave order phase diagram involves two parameters: in addition to the parameters, Kondo singlets also appear as the competing Kondo coupling JK, there is also G which measures the variables dual to the AF order. In this basis, the conduction degree of the quantum fluctuations of the local-moment electrons acquire a Berry phase through their coupling to the magnetism. The vertical axis reflects tuning geometrical hedgehog configurations of the Néel order, which cancels the frustration or dimensionality. The Kondo coupling, depicted Berry phase of the local moments. These results demonstrate as the horizontal axis, is taken to be dimensionless with the the competition between the Kondo-singlet formation and conduction-electron bandwidth W as the normalization factor. spin-Peierls order when the AF order is suppressed, in a way The global phase diagram itself is a two-dimensional that is compatible with the global phase diagram discussed projection of a multi-dimensional phase diagram. In partic- earlier. ular, we have considered the case with a fixed I=W that is considerably smaller than 1. In addition, we have fixed, x, the 6. Antiferromagnetic Order: Kondo Destruction vs number of conduction electrons per site, to some non-integer Spin-Density-Wave Order value. We have so far emphasized that the stability of the AFS There are three sequences of phase transitions from the phase has been derived based on an (asymptotically exact) AFS phase to the PL phase. Trajectory I represents a direct RG analysis. The exact marginality of the Kondo coupling transition involving Kondo destruction, and corresponds to inside the AF order is to be contrasted with its marginal the local QCP. Trajectory II involves an intermediate AFL relevance in the paramagnetic case. Because the effective phase, which represents the SDW order of the PL phase. Kondo coupling does not grow, the system no longer flows to There is a Kondo-destruction transition inside the AF order, the strong-coupling Kondo fixed point; in the ground state, while the AF to non-magnetic transition is of the SDW type. the static Kondo singlet has zero amplitude. However, the Trajectory III involves an intermediate PS phase, which marginal nature of the effective Kondo coupling also implies could involve non-magnetic order such as a valence-bond that the Kondo coupling influences the properties at non-zero solid. energies. In other words, a dynamical Kondo effect operates. 061005-5 ©2014 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Rice University on 06/01/17

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Fig. 7. Dynamical Kondo effect in the AFS phase, for various values of 0 41) I=TK on the ordered side.

This analysis complements the results from the EDMFT studies in the ordered state. Figure 7 shows the local dynamical spin susceptibility as a function of frequency in 41) the AFS state. Its increase as ¤ is tuned towards the QCP reflects the growth of the dynamical Kondo effect. When ¤ reaches c, the local dynamics in the ordered state match the quantum critical behavior determined from the EDMFT Fig. 8. (Color online) AFL and AFL2 phases in the SDW portion of the studies in the absence of order. This demonstrates the phase diagram. Uk , with ¼ 1; ...; 4, are the heavy-fermion bands in the importance of the dynamical Kondo effect in ensuring the presence of a staggered field associated with a staggered magnetization. second-order nature of the zero-temperature transition. The same conclusions also emerge in several other studies.43,44) Indeed, the dynamical Kondo effect is important for the destruction in particular, have been playing a central role stability of the AFS phase. It allows the gain of the Kondo in the modern studies of heavy fermion magnetism and exchange energy even in the absence of the static Kondo superconductivity. Here we briefly consider the salient singlet formation. This point is important to the under- properties of the theory that have either been compared to standing of the results from several variational Quantum known experiments, or represent predictions that have been Monte Carlo studies.63–65) These studies used a variational tested by subsequent experiments. More extensive discus- wavefunction for the AFS phase that not only sets the static sions may be found in Refs. 67 and 68. Kondo amplitude to zero, but also disallows any Kondo We have already mentioned the strong evidence8,10–13) for fluctuations at finite energies. As such, it cannot energetically local quantum criticality from the heavy-fermion compounds compete against the AFL phase, defined in terms of a CeCu6xAux, YbRh2Si2, and CeRhIn5. This concerns the variational wavefunction with a static Kondo-singlet ampli- anomalous dynamical scaling, an extra energy scale and a tude. In this way, the approach does not adequately capture sudden jump of the Fermi surface. Additional evidence has 69–71) the dynamical competition between the RKKY and Kondo come from transport measurements in CeRhIn5 and 72) interactions. Correspondingly, it is difficult to stablize the NMR studies of YbRh2Si2. AFS phase. Instead, these studies would only allow the The proposed global phase diagram has helped understand multiple AF ground states with a Lifshtiz transition inside a surprisingly rich zero-temperature phase diagram of the 73,74) the SDW AFL phase. In the example shown in Fig. 8, this Ir- and Co-substituted YbRh2Si2. Likewise, it may also corresponds to going from the usual AFL phase for small AF provide a means to understand the variety of quantum order parameter, with co-existing electron and hole pockets, phase transitions under the multiple tuning parameters in 75) 76) to the AFL2 phase for larger AF order parameter, in which CeCu6xAux and CeRhIn5. the hole pocket has disappeared. Interestingly, the standard The global phase diagram has suggested that increasing DMFT approach likewise over-emphasizes the Kondo dimensionality tunes the occurrence of Kondo destruction coupling, because the RKKY interactions do not appear in from at the onset of AF order to inside the ordered region, the dynamical equations [the bosonic bath in Eq. (7) is absent which is consistent with the recent measurements in 77) in DMFT]. The approach therefore reduces the regime of Ce3Pd20Si6. 66) stability of the AFS phase. In the terminology of Fig. 6, it Finally, it also suggests that heavy-fermion materials with captures the type II transition but misses the type I transition lattices that host geometrically-frustrated magnetism would (or, for that matter, the type III transition as well). be particularly instructive in exploring the upper portion of the phase diagram, i.e., the region where the local-moment 7. Experiments on Quantum Critical Heavy Fermions component contains especially strong quantum fluctuations. Quantum phase transitions in general, and Kondo This has provided the motivation for recent studies of heavy- 061005-6 ©2014 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Rice University on 06/01/17

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(a) (b)

Fig. 9. (Color online) (a) Dynamical spin structure factor in the normal and superconducting states of CeCu2Si2, from Ref. 82. (b)–(d) Interpretation of the large exchange energy gain in terms of a nonzero but small Kondo-destruction energy scale Eloc, as introduced in Refs. 82 and 83 and described in the main text.

– 78) 79) 80) 90) fermion metals on Shastry Sutherland, Kagome, fcc, SDW, !=T, scaling form; this suggests that Eloc at the QCP and triangular81) lattices. is on the order of 1 K. In recent years, STM studies have been carried out in a number of heavy-fermion systems.91–93) We 8. Implications for Superconductivity anticipate that the kinetic energy loss can be estimated Unconventional superconductivity often arises in the through STM measurements of the single-particle spectral vicinity of magnetic instabilities.84) At the same time, the function both in the normal and superconducting states. superconducting phases found in rare earth intermetallic The evidence for the Kondo destruction quantum criticality compounds have rich and diverse properties. It is therefore determining superconductivity is the most direct in CeRhIn5 natural to suspect that the global phase diagram for the heavy (Refs. 11 and 69). In this compound under pressure, AF order fermions with its various magnetic transitions will have is weakened and eventually gives way to superconductiv- 94) implications for the emergence of superconductivity. That ity, as shown in Fig. 10(a); Tc 2:3 K is high in that it is antiferromagnetic spin fluctuations promote unconventional about 10% of the bare Kondo temperature. Applying a superconductivity has been suggested85,86) soon after the magnetic field suppresses superconductivity and uncovers an discovery of unconventional superconductivity in CeCu2Si2 AF QCP, as seen in Fig. 10(b). Across this QCP, the Fermi by Steglich.87) Inelastic neutron scattering intensity has been surface experiences a sudden jump [Fig. 10(c)]. This measured both in the normal and superconducting states of provides evidence for Kondo destruction, which is corrobo- CeCu2Si2 near quantum criticality, as shown in Fig. 9(a). The rated by the divergence of the effective mass [Fig. 10(d)]. All results have been used to estimate the gain in exchange these suggest that the high-temperature superconductivity in 69) energy across the transition, which is an order of magnitude CeRhIn5 originates from local quantum criticality. larger than the condensation energy.82) (Related conclusion 88) 9. Spin–Orbit Coupling and Topological Phases has also been reached in CeCoIn5. ) While establishing that the magnetism drives the formation of superconductivity, it 9.1 Global phase diagram of Kondo insulators also implies that a correspondingly large kinetic energy is lost A generalization to the commensurate conduction-electron across the superconducting transition. The latter has been filling of x ¼ 1 leads to the corresponding global phase interpreted in terms of a Kondo-destruction energy scale diagram for Kondo insulators,52) shown in Fig. 11. As 67) Eloc being nonzero but quite small: as illustrated in Figs. 9(c) discussed elsewhere, various material families could be and 9(d), this allows the further reduction of the Kondo- considered as candidates for inducing transitions between singlet amplitude by superconductivity to transfer a sub- these different phases. stantial amount of the Kondo-resonance spectral weight to higher energies, causing a large loss of the Kondo screening 9.2 Topological phases and their transitions to magnetic energy which is counted as a part of the kinetic energy. and Kondo states Indeed, while considerable evidence exists that the low- SmB6 has been the focus of many renewed experi- energy and low-temperature properties are consistent with the ments.95,96) These have followed the suggestion that the SDW type of QCP,89,90) the dynamics above a relatively low strong spin–orbit coupling of the 4f-electrons induce non- temperature (³1 K) appears to be consistent with the non- trivial topology in the heavy-fermion bandstructure, which 061005-7 ©2014 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Rice University on 06/01/17

J. Phys. Soc. Jpn. 83, 061005 (2014) Special Topics Q. Si et al.

(a) (b)

Fig. 10. (Color online) Phase diagram of CeRhIn5 in the T–p plane at zero field (a) and in the B–p plane close to zero temperature (b); figures adapted from Ref. 69. Also shown are the dHvA measurements as a function of pressure, demonstrating a sudden jump of the dHvA frequencies (c), which indicates a corresponding jump of the Fermi surface, and a tendency of divergence in the effective mass (d); figures adapted from Ref. 11.

Given the stoichiometric nature of the system, it would then be natural to suggest that superconductivity will appear in a similar pressure range as a consequence of quantum criticality. Another transition at a Kondo-insulator ﬁlling is between 100) aPS phase and a KI phase. This has been studied in a Kondo lattice model supplemented by a spin–orbit coupling (SOC) for the conduction electrons:

H ¼ HKL þ HsocðcÞ: ð15Þ The details of the Hamiltonian are given in Ref. 100. The Fig. 11. (Color online) Global phase diagram of Kondo insulators,52) and spin–orbit coupling term in this Hamiltonian induces a representative materials that may be tuned through various transitions.67) topological insulator state for the conduction electrons. Figure adapted from Ref. 67. Because of the TI gap of the conduction electrons, the Kondo coupling JK must be larger than a non-zero threshold 97) c turn the Kondo insulator into a topological insulator (TI). value JK in order to reach a Kondo insulator phase. A large-N At the present time, there is considerable evidence for surface analysis yields a continuous transition between the TI and states in SmB6, and whether these are the boundary states of Kondo insulator phases. It is likely that magnetic order will the bulk TI phase remains to be established. Still, it is also interplay with these phases, and studying this effect in instructive to consider SmB6 as a case in which the bulk KI the model should be very instructive. gap can be closed by the application of pressure. When that happens, the system becomes metallic and magnetically 9.3 Heavy fermion interfaces ordered,98) making the trajectory of phase transition to be Such consideration of the spin–orbit coupling also likely along the dashed line shown in Fig. 11. suggests the intriguing possibility of new properties at the With this in mind, it is intriguing to note the transport interface of heavy-fermion heterostructures. Because of the evidence for non-Fermi liquid behavior in SmB6 under a broken inversion symmetry at the interface, the heavy 99) pressure of about 4 GPa, in the transition regime. This electrons in the interface layerP should contain an extra SOC suggests that the zero-temperature transition from the KI of the Rashba type: Hsoc ¼ k Vsocðn kÞsðkÞ, where k is phase to the AFS is (close to being) second order, and the the wavevector, Vsoc the spin–orbit coupling, n the unit vector associated QCP underlies the non-Fermi liquid behavior. perpendicular to the interface, and sðkÞ the spin of the 061005-8 ©2014 The Physical Society of Japan J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by Rice University on 06/01/17

J. Phys. Soc. Jpn. 83, 061005 (2014) Special Topics Q. Si et al. electrons with wavevector k. For the oxide heterostructures, [email protected] 1) Special issue: Quantum Phase Transitions, J. Low Temp. Phys. 161, a Rashba-type SOC with Vsoc on the order of 5 meV has 101,102) 1 (2010). been demonstrated. Such a SOC energy scale will be 2) Q. Si and F. Steglich, Science 329, 1161 (2010). competitive against the heavy-fermion energy scales, raising 3) P. Pfeuty, Ann. Phys. (N.Y.) 57, 79 (1970). the possibility for topologically non-trivial superconducting 4) A. P. Young, J. Phys. C 8, L309 (1975). or insulating states at such heavy-fermion interfaces. Heavy- 5) J. Hertz, Phys. Rev. B 14, 1165 (1976). fermion heterostructures appear to be quite realistic to study. 6) Q. Si, S. Rabello, K. Ingersent, and J. Smith, Nature 413, 804 (2001). 7) P. Coleman, C. Pépin, Q. Si, and R. Ramazashvili, J. Phys.: Condens. 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(1991) degree from the University of Thompson, F. Steglich, and Q. Si, arXiv:1308.0294. Chicago. He did his postdoctoral works (1991–1995) 77) J. Custers, K. Lorenzer, M. Müller, A. Prokofiev, A. Sidorenko, H. at Rutgers University and University of Illinois at Winkler, A. M. Strydom, Y. Shimura, T. Sakakibara, R. Yu, Q. Si, and Urbana—Champaign. In 1995 he joined the faculty S. Paschen, Nat. Mater. 11, 189 (2012). of Rice University, where he is the Harry C. and 78) M. S. Kim and M. C. Aronson, Phys. Rev. Lett. 110, 017201 (2013). Olga K. Wiess Professor of Physics. His research is 79) V. Fritsch, N. Bagrets, G. Goll, W. Kittler, M. J. Wolf, K. Grube, C.-L. in the field of theoretical condensed matter physics, Huang, and H. v. Löhneysen, Phys. Rev. B 89, 054416 (2014). with a focus on strongly correlated electron systems. Specific research 80) E. D. Mun, S. L. Bud’ko, C. Martin, H. Kim, M. A. Tanatar, J.-H. subjects have included quantum criticality, non-Fermi liquid physics, heavy Park, T. Murphy, G. M. Schmiedeshoff, N. Dilley, R. Prozorov, and fermion phenomena, high temperature cuprate and iron-pnictide super- P. C. Canfield, Phys. Rev. B 87, 075120 (2013). conductivity, and mesoscopic and disordered electronic systems. 81) D. D. Khalyavin, D. T. Adroja, P. Manuel, A. Daoud-Aladine, M. Kosaka, K. Kondo, K. A. McEwen, J. H. Pixley, and Q. Si, Phys. Rev. Jedediah H. Pixley was born in Baltimore Mary- B 87, 220406 (2013). land, in the United States of America in 1985. He 82) O. Stockert, J. Arndt, E. Faulhaber, C. Geibel, H. S. Jeevan, S. received his B.A. in pure mathematics and B.S. in Kirchner, M. Loewenhaupt, K. Schmalzl, W. Schmidt, Q. Si, and F. physics (2008) from the University of California Steglich, Nat. Phys. 7, 119 (2011). Santa Cruz, graduating with the highest honors for 83) O. Stockert, S. Kirchner, F. Steglich, and Q. Si, J. Phys. Soc. Jpn. 81, both degrees. He is currently a Ph. D. candidate at 011001 (2012). Rice University (expected to be conferred May of 84) N. Mathur, F. Grosche, S. Julian, I. Walker, D. Freye, R. 2014), and will be moving to the Condensed Matter Haselwimmer, and G. Lonzarich, Nature 394, 39 (1998). Theory Center at the University of Maryland as a 85) D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B 34, 8190 Postdoctoral Fellow in the Fall of 2014. He has (1986). worked on the theory of classical phase transitions in disordered magnets and 86) K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 relaxation dynamics of polymers. His current research focuses on quantum (1986). criticality and unconventional superconductivity in heavy fermion metals as 87) F. Steglich, J. Aarts, C. Bredl, W. Lieke, D. Meschede, W. Franz, and well as frustrated quantum magnetism in both insulating and metallic H. Schäfer, Phys. Rev. Lett. 43, 1892 (1979). systems. 88) C. Stock, C. Broholm, J. Hudis, H. J. Kang, and C. Petrovic, Phys.

J. Phys. Soc. Jpn. 83, 061005 (2014) Special Topics Q. Si et al.

Emilian Marius Nica was born in Slatina, Olt Rong Yu was born in Beijing, China in 1975. He County, Romania in 1986. He obtained his B.Sc. obtained his B.S. degree from Peking University in (2009) from Texas A&M University, his M.Sc. 1998, M.S. degree from Tsinghua University in (2013) from Rice University and is currently 2001, and Ph. D. degree from University of South- pursuing his D.Sc. degree from the same institution. ern California in 2007. He was a postdoctoral He has worked on the theories of quantum criticality research associate at University of Tennessee, Knox- in heavy-fermion compounds and of strong electron ville (2007–2009) and at Rice University correlations in unconventional superconductors. (2009–2013). Since 2013, he has been an associate professor at Department of Physics, Remin Univer- sity of China. He has been working on theory of correlated electronic systems. Current main areas of his research includes Seiji Yamamoto received bachelor degrees in phase transitions in heavy fermion systems, frustration and disorder effects in Physics and Electrical Engineering from Stanford quantum magnets, superconductivity and correlation effects in iron-based University, and MS and Ph. D. degrees in Physics superconductors. from Rice University. His research publications focus on renormalization group calculations of Stefan Kirchner was born in Fulda, Germany in ﬁ effective eld theories for certain classes of heavy 1971. He studied physics at the State University of fermion compounds. As a post-doc at the National New York, U.S.A. and the University of Würzburg, High Magnetic Field Lab, Seiji worked on a Germany. After completion of his diploma, he theoretical method to detect 3D non-abelian anyons moved to Karlsruhe to work on his Ph. D. He at the boundary between a topological insulator and received his Ph. D. from the Technical University of a superconductor. Karlsruhe (now Karlsruhe Institute of Technology). From 2003 to 2009 he worked as a research Pallab Goswami was born in Kolkata, India in associate at Rice University in Houston, U.S.A. 1978. He obtained his B. Sc. (2000), M. Sc. (2002), Since 2009 he is junior research group leader of the and Ph. D. (2008) degrees in Physics from Jadavpur Max Planck Institute for Physics of Complex Systems and Chemical Physics University, Indian Institute of Technology Kanpur, of Solids in Dresden, Germany. He works on the theoretical description of and University of California Los Angeles respec- strongly interacting systems, in particular dilute and dense Kondo systems tively. Subsequently he has been a Postdoctoral with a recent emphasis on quantum phase transitions and the emergence of – Fellow at Rice University (2008 2011) and at novel states associated with quantum criticality. National High Magnetic Field Laboratory, Tallahas- see (2011 to present). He has worked on competing ordered states and their inﬂuence on quantum phase transitions in various strongly correlated and disordered systems. Currently his research interest is focused on unconventional superconductivity, topological states of matter and role of topological defects in inducing competing order and unconventional quantum phase transitions.